Evaluates informed hypotheses on multinomial parameters. These hypotheses can contain (a mixture of) inequality constraints, equality constraints, and free parameters. Informed hypothesis \(H_r\) states that category proportions obey the particular constraint. \(H_r\) can be tested against the encompassing hypothesis \(H_e\) or the null hypothesis \(H_0\). Encompassing hypothesis \(H_e\) states that category proportions are free to vary. Null hypothesis \(H_0\) states that category proportions are exactly equal.
mult_bf_informed(
x,
Hr,
a = rep(1, length(x)),
factor_levels = NULL,
cred_level = 0.95,
niter = 5000,
bf_type = "LogBFer",
seed = NULL,
maxiter = 1000,
nburnin = niter * 0.05
)List consisting of the following elements
$bf_listgives an overview of the Bayes factor analysis:
bf_type: string. Contains Bayes factor type as specified by the user
bf: data.frame. Contains Bayes factors for all Bayes factor types
error_measures: data.frame. Contains for the overall Bayes factor
the approximate relative mean-squared error re2, the approximate coefficient of variation cv, and the approximate percentage error percentage
logBFe_equalities: data.frame. Lists the log Bayes factors for all independent equality constrained hypotheses
logBFe_inequalities: data.frame. Lists the log Bayes factor for all independent inequality constrained hypotheses
$cred_levelnumeric. User specified credible interval
$restrictionslist that encodes informed hypothesis for each independent restriction:
full_model: list containing the hypothesis, parameter names, data and prior specifications for the full model.
equality_constraints: list containing the hypothesis, parameter names, data and prior specifications for each equality constrained hypothesis.
inequality_constraints: list containing the hypothesis, parameter names, data and prior specifications for each inequality constrained hypothesis.
In addition, in nr_mult_equal and nr_mult_free encodes which and how many parameters are
equality constraint or free, in boundaries includes the boundaries of each parameter, in nineq_per_hyp states the number of inequality constraint
parameters per independent inequality constrained hypothesis, and in direction states the direction of
the inequality constraint.
$bridge_outputlist containing output from bridge sampling function:
eval: list containing the log prior or posterior evaluations
(q11) and the log proposal evaluations (q12) for the prior or posterior samples,
as well as the log prior or posterior evaluations (q21) and the log proposal evaluations (q22)
for the samples from the proposal distribution
niter: number of iterations of the iterative updating scheme
logml: estimate of log marginal likelihood
hyp: evaluated inequality constrained hypothesis
error_measures: list containing in re2 the approximate
relative mean-squared error for the marginal likelihood estimate, in cv the approximate
coefficient of variation for the marginal likelihood estimate (assumes that bridge estimate is unbiased), and
in percentage the approximate percentage error of the marginal likelihood estimate
$sampleslist containing a list for prior samples and a list
of posterior samples from truncated distributions which were used to evaluate inequality constraints.
Prior and posterior samples of independent inequality constraints are again saved
in separate lists. Samples are stored as matrix of dimension nsamples x nparams.
numeric. Vector with data
string or character. Encodes the user specified informed hypothesis. Use either specified factor_levels
or indices to refer to parameters. See ``Note'' section for details on how to formulate informed hypotheses
numeric. Vector with concentration parameters of Dirichlet distribution. Must be the same length as x. Default sets all concentration parameters to 1
character. Vector with category names. Must be the same length as x
numeric. Credible interval for the posterior point estimates. Must be a single number between 0 and 1
numeric. Vector with number of samples to be drawn from truncated distribution
character. The Bayes factor type. When the informed hypothesis is compared to the encompassing hypothesis,
the Bayes factor type can be LogBFer, BFer, or BFre. When the informed hypothesis is compared to the null hypothesis,
the Bayes factor type can be LogBFr0, BF0r, or BFr0. Default is LogBFer
numeric. Sets the seed for reproducible pseudo-random number generation
numeric. Maximum number of iterations for the iterative updating scheme used in the bridge sampling routine. Default is 1,000 to avoid infinite loops
numeric. A single value specifying the number of burn-in samples when drawing from the truncated distribution. Minimum number of burn-in samples is 10. Default is 5% of the number of samples. Burn-in samples are removed automatically after the sampling.
The model assumes that data follow a multinomial distribution and assigns a Dirichlet distribution as prior for the model parameters (i.e., underlying category proportions). That is: $$x ~ Multinomial(N, \theta)$$ $$\theta ~ Dirichlet(\alpha)$$
damien2001samplingmultibridge
gronau2017tutorialmultibridge
fruhwirth2004estimatingmultibridge
sarafoglou2020evaluatingPreprintmultibridge
Other functions to evaluate informed hypotheses:
binom_bf_equality(),
binom_bf_inequality(),
binom_bf_informed(),
mult_bf_equality(),
mult_bf_inequality()
# data
x <- c(3, 4, 10, 11, 7, 30)
# priors
a <- c(1, 1, 1, 1, 1, 1)
# restricted hypothesis
factor_levels <- c('theta1', 'theta2', 'theta3', 'theta4', 'theta5',
'theta6')
Hr <- c('theta1', '<', 'theta2', '&', 'theta3', '=', 'theta4',
',', 'theta5', '<', 'theta6')
output_total <- mult_bf_informed(x, Hr, a, factor_levels, seed=2020, niter=2e3)
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